General Jump-Diffusion Setting

In the preceding chapter we showed that in a model with Gaussian diffusion factors the asset allocation problem reduces, via the change of measure technique, to a controlled diffusion problem in the factor process, even though there are jumps in the asset price model. The problem can be handled by classical methods of stochastic control, and the result is C1,2 regularity of the value function. In this chapter we address the same problem, but in a model with jumps in the factors as well as the assets. Of course, the factor jumps will no longer disappear under change of measure, so our equivalent stochastic control problem will lead to an HJB equation in the form of a PIDE (partial integro-differential equation), a PDE with a non-local term. The reader can consult Øksendal and Sulem (2005) and Pham (2010) for excellent introductions to problems of this type. Our motivation for including jumps in the factors is that they enable us to represent the effect of discrete events, such as changes in official interest rates, on asset prices. At the same time we greatly expand the scope of the model in this chapter beyond that in Chapter 7 by including investment constraints, stochastic volatility effects and nonlinear dependence of the asset model parameters on the factors. Nonetheless, the main conclusion of the chapter is that (under conditions we state), C1,2 regularity of the value function is maintained…